Tagged Questions

Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic ...

I was thinking about the following "obvious" construction the other day. Let
$$
\mathbb{S}^{\infty}_{2}=\bigcup_{i=1}^{\infty}\mathbb{S}^{2i}
$$
Then because we know that $\chi(\mathbb{S}^{2i})=2$ for ...

Is it true that the space of Fredholm operators on a separable Hilbert space is the classifying space for K-theory in the category of paracompact spaces?
Everyone quotes the theorem of Atiyah-Janich ...

I have a question concerning the so called Local Index Formula by Connes in noncommutative geometry. First issue:
why it is called Index Formula?
I spoke to one person about this and he gave me the ...

I read a physics paper, of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a ...

Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...

The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...

I have a very stupid and non-specific question, as follows. And let me know if I am asking in a wrong way.
We known that, for instance, if one is interested in computing dimension of moduli space of ...

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...

This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne.
Let $M$ be a compact Riemannian manifold without boundary and $P\rightarrow M$ ...

I have the following, probably very elementary question: Let $Cl^{p,q}$ be the Clifford algebra on generators $e_i$, $i=1, \ldots, p+q$
with $e_i e_j = -e_j e_i$ and $e_{i}^{2}=-1$ for $i=1,\ldots,p$, ...

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...

Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...

Hi,
I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn.
I do not understand why the analytic index map ...